Thymic Presentation of Autoantigens and the Efficiency of Negative Selection

Thymic Presentation of Autoantigens and the Efficiency of Negative Selection

HUGO A. VAN DEN BERG*^,†and CARMEN MOLINA-PARI´S^‡,§

Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (Received 11 March 2003; In final form 21 June 2003)

Antigen recognition by the adaptive cellular immune system is based on a diverse repertoire of antigen receptors. Since this repertoire is formed by genetic recombination, a number of receptors are autoreactive by chance, giving rise to the threat of autoimmune disease. Potentially autoreactive T lymphocytes (T cells) are rendered ineffective by various tolerance mechanisms. One of these mechanisms is negative selection, the deletion from the repertoire of immature autoreactive T cells in the thymus. The present paper shows how to assess the contribution made by negative selection relative to other tolerisation mechanisms by deducing the impact of negative selection on the T cell repertoire from the statistics of autoantigen presentation in the thymus.

Keywords: T cell tolerance; Negative selection; Autoimmunity; Large deviations theory

INTRODUCTION

T cell tolerance is a complex phenomenon which comprises both central and peripheral tolerance: lympho- cytes acquire the former while they are developing in the primary lymphoid tissues, the latter as mature cells recirculating through the secondary lymphoid tissues (Janeway and Travers, 1997). The present paper shows how to resolve T cell tolerance into its central and peripheral contributions by a comparative analysis of the statistics of autoantigen presentation in the primary and secondary lymphoid tissues.

The adaptive cellular immune system recognises pathogenic antigens by means of the T cell antigen receptor (TCR), which interacts with peptide antigens displayed on the surface of antigen presenting cells (APCs) by glycoproteins belonging to the major histo- compatibility complex (MHC) (Janeway and Travers, 1997). The immune system contains millions of distinct TCR molecules, formed by random rearrangement of the gene segments encoding the region of the TCR molecule that interacts with the peptide-MHC (pMHC) complex (Janeway and Travers, 1997; Arstila et al., 1999). Each T cell expresses one specific TCR species or clonotype, unique to the T cell and the clone to which it belongs.

Since the generation of TCR clonotypes proceeds at random, some of them inevitably are autoreactive, that is, their TCR molecule recognises one or more antigens derived from the body’s own proteins (autoantigens). Such autoreactive clones are kept in check by various tolerisation mechanisms which act to prevent their activation and concomitant autoimmune disease. Follow- ing recognition of autoantigens presented by a tolerising APC, an autoreactive T cell may be eliminated, or forced to reduce its responsiveness, or to become a suppressor cell (Webbet al., 1990; Antoniaet al., 1995; Smithet al., 1997; Seddon and Mason, 1999; Roncarolo and Levings, 2000; Steinmanet al., 2000; Hawigeret al., 2001).

The T cell repertoire undergoes central tolerisation in the thymus, where numerous immature T cells (thymo- cytes) that recognise autoantigens are induced to undergo apoptosis (Kappler et al., 1987; Kisielow et al., 1988;

Surh and Sprent, 1994). While this process of negative selection is generally thought to prevent maturation of many autoreactive T cells, it seems likely that some degree of autoreactivity remains in the mature repertoire (Tanchotet al., 1997; Bouneaudet al., 2000; Visseret al., 2000; Zippelius et al., 2002). Low-avidity TCR interaction with autoantigens accounts for part of this residual autoreactivity (Bouneaudet al., 2000; Herna´ndez et al., 2000; Nugent et al., 2000; Visser et al., 2000).

ISSN 1027-3662 print/ISSN 1607-8578 onlineq2003 Taylor & Francis Ltd DOI: 10.1080/102736620310001604910

*Corresponding author. Tel.:þ44-2476-523698. Fax:þ44-2476-524182. E-mail: hugo@maths.warwick.ac.uk

†Supported by the Wellcome Trust.

‡Supported by the EPSRC.

§Present address: Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT.

Moreover, negative selection would fail to abolish autoreactivity completely if negatively selecting cells in the thymus could only present constitutive and lineage-specific peptides, as opposed to tissue-specific and sequestered peptides, a traditional argument no longer deemed to be strictly correct (Klein et al., 1998;

Sospredaet al., 1998).

A deeper reason why autoreactivity cannot be completely eliminated is that it becomes virtually impossible for negative selection to abolish autoreactivity entirely if the number of autoantigens recognised by a pre- selection thymocyte is much larger than 1, which is likely to be the case (Hogquistet al., 1997; Mason, 2001). Some degree of residual autoreactivity in the mature T cell repertoire thus seems inevitable, which suggests that negative selection should be regarded as modifying rather than eliminating autoreactivity in the TCR repertoire.

The autoreactivity of an individual TCR clonotype involves more than merely the number of autoantigens recognised by its TCR molecule. Equally important for the immunogenic potential of an autoantigen are (i) its ubiquity, that is, the frequency with which a mature recirculating T cell will encounter the autoantigen as it visits secondary lymphoid tissues throughout the body (Janeway and Travers, 1997; Klein et al., 1998) and (ii) its presentation level, that is, its copy number on the surface of the APCs (Akkaraju et al., 1997; Byers and Lindahl, 1999; Kurts et al., 1999; Morgan et al., 1999;

Sebza et al., 1999; Reay et al., 2000). Moreover, a thymocyte need not encounter a given autoantigen with the same ubiquity or presentation level as a mature T cell does. For instance, thymic negatively selecting cells may not be able to present certain autoantigens derived from proteins with non-constitutive (inducible) or tissue- restricted expression, and these autoantigens would have zero ubiquity in the thymus. However, a growing body of experimental evidence indicates to the contrary that negatively selecting cells are able to express and present rare peripheral antigens that are restricted to specific cell types in the periphery (Antonia et al., 1995; Fritz and Zhao, 1996; Smithet al., 1997; Farr and Rudensky, 1998;

Hanahan, 1998; Kleinet al., 1998; Sospredaet al., 1998).

Such ectopic thymic expression of peripheral autoantigens would seem to broaden considerably the scope and efficiency of central tolerance.

The analysis of the autoreactivity of the T cell repertoire is technically challenging because of the presence of two kinds of variability: interclonotypic variability due to random differences between the autoreactivity of various TCR clonotypes and intraclonotypic variability due to fluctuations in the autoantigens presented on various APCs. In a sense, the repertoire is a “distribution of distributions”: each clonotype has its own distribution law from which it samples as it registers signals on subsequent APCs, and then the whole repertoire is a collection of samples from a distribution of laws. Tolerance can only affect the latter distribution, not the law itself, since it is intrinsic to the TCR’s molecular structure.

An earlier paper (van den Berget al., 2001) analysed the simple extreme case where self peptides encountered on APCs in the secondary lymphoid tissues are either constitutive (present on all APCs) or exceedingly rare (found on a very small fraction of APCs). All clonotypes then share the same law up to an additive term, and the TCR repertoire is fully characterised by the distribution of the latter term over the clones. However, as noted in that paper, on this simplification one cannot deal with autoantigens of intermediate ubiquity, differences in self presentation statistics in the thymus as opposed to the periphery, and the distinct contributions made, respect- ively, by central and peripheral tolerance. The purpose of the present paper is to carry out a more general analysis which allows these issues to be addressed. Thus, the theory developed in this paper describes precisely how patterns of thymic presentation shape the statistical structure of the mature T cell repertoire. In particular, the present theory shows how data on the thymic ubiquities and presentation levels of autoantigens, ranging from very rare to constitutive, can be used to characterise how negative selection modifies the reper- toire, which will enable immunologists to delineate the relative contributions of central versus peripheral tolerance.

Organisation of the paper: The second section develops a large deviations description of the stochastic variability of T cell stimulation due to autoantigens, broken down into interclonotypic variability due to random differences between the autoreactivity of various TCR clonotypes and intraclonotypic variability due to fluctuations in the autoantigens presented on various APCs. The third section presents the main results. The repertoire structure induced by central tolerance is related explicitly to thymic autoantigen presentation statistics.

Furthermore, the efficiency of negative selection is precisely characterised. The fourth section outlines an application of the theory, showing by means of two examples how comparison of central and peripheral auto- antigen presentation can be used to elucidate the role of negative selection. Notation used throughout is summar- ised in Table I; additional notation is always explained locally.

FLUCTUATIONS IN T CELL STIMULATION BY SELF

Upon conjugation with an antigen-presenting cell, a T cell registers a signal through its TCR molecules due to the peptides presented on the MHC molecules on the APC surface. This TCR signal will be represented as a weighed sum over the contributions due to the various pMHC species, with weighing factors corresponding to the densities of the various pMHC species on the APC.

The T cell will be assumed to respond when the TCR signal exceeds a cellular threshold (Viola and Lanzavecchia, 1996).

Autorecognition of various autoantigens by a given clonotype can no longer be treated as statistically independent following negative selection: if A and B are two arbitrary autoantigens, the probability that a mature T cell, chosen at random, recognises A given that it recognises B is lower than the unconditional probability that a random mature T cell recognises A (both probabilities are of course generally lower than the probability that A is recognised by a thymocyte chosen at random prior to negative selection). Since the total number of peripheral autoantigens, rare as well as abundant, is very large, a direct attack on the correlations induced by negative selection using iterative conditioning quickly becomes very cumbersome. Fortunately, a more tractable way to deal with the correlations is offered by large deviations techniques, which exploit the vastness of the number of autoantigens.

Variability of the TCR signal due to recognition of autoantigens has two major sources: one is interclonotypic variability, which arises because different TCR molecules recognise different antigens, and another is intraclono- typic variability due to random fluctuations in antigen presentation. To represent both kinds of fluctuations in

a manner that is amenable to large deviations techniques, the autoantigens will be lumped into presentation components.

T Cell Activation

The TCR signal not only depends on the TCR clonotypei, but also on the copy numbers of the various pMHC species on the APC. The latter can be represented as an antigen presentation profile (APP)

z^def¼ {z1;z2;. . .}

wherezjdenotes the relative presentation level of pMHC speciesj, defined by

zj

def¼ Zj=MT

whereM_Tis the number of pMHC molecules capable of binding to the TCR at hand andZ_jis the number of pMHC molecules of species j; the relative presentation levelzj

sums to 1 over the pMHC speciesj.

Let w_iz denote the TCR signal registered by a T cell of TCR clonotype i conjugated with an APC presenting

TABLE I Notation

Symbol Interpretation

i index for T cell clonotypes

j index for pMHC species

k index for self presentation components

N number of self pMHC species with which a given TCR can interact, with or without productive recognition Nk number of self pMHC species belonging to componentk

N^ik number of self pMHC species recognized by clonotypei, belonging to componentk

L^i clonotype law, L^i¼ fL^ikg^K_k¼1whereL^ik¼N^ik=Nis the autorecognition frequency of cloneifor componentk LN set of all possible clonotype laws

K number of self presentation components

Kk j[Kkiff pMHC speciesjbelongs to componentk;jKkj ¼Nk

m probability that a random TCR chosen recognizes a pMHC species that is chosen at random from among the pMHCs with which the TCR can interact

z APP,z¼ fz_jg^N

j¼1wherezjis relative presentation level of pMHC speciesj zj relative presentation level of pMHC speciesj, averaged over APPs Ijz indicator which equals 1 whenzj.0 in APPz, and 0 otherwise

uj ubiquity, probability that pMHC speciesjappears in a randomly chosen APPzuj¼P(Ijz¼ 1) ndiv presentation diversity,ndiv¼1/PN

j¼1z²_j

nkz number of self pMHC species withzj.0in APPz, belonging to componentk

^

nikz number of self pMHC species withzj.0in APPz, belonging to componentkand recognized by clonotypei wiz TCR signal elicited in clonotypeiby APPz

wij TCR signal elicited in clonotypeiby a pMHC complex of speciesj wact cellular activation threshold

wthy thymic negative selection threshold

wi TCR signal due to self in clonotypei, averaged over APPs s²_APP variance over APPs of TCR signal due to self in clonotypei p component partitioning,p¼ fpkg^K_k¼1wherepk ¼Nk/N r component presentation,r¼ frkg^K_k¼1whererk= limN!1P

j[Kkzj

u component ubiquities,u¼ fukg^K_k¼1

u mean effective ubiquity,u¼ PK k¼1r²_kp²¹_k . PK

k¼1r²_kðpkukÞ²¹

rj presentation propensity of pMHC speciesj rTz total presentation propensity,rTz¼PN

jrjIjz

m number of rounds of negative selection in the thymus

mcrit critical number of rounds of negative selection,mcrit ¼ ð12uÞ= u Psurv probability that a thymocyte survives negative selection

I,I,J large deviations rate functions

APP, antigen presentation profile; TCR, T cell antigen receptor; pMHC, major histocompatibility complex molecule, presenting a peptide; pAPC, professional antigen- presenting cell.

A prime indicates statistics for peripheral pAPCs; unprimed statistics refer to cells effecting negative selection. Not included are notations used and explained locally.

See “Discussion” section for further remarks on the interpretation ofP_surv.

a pMHC ensemblez. Thesummation hypothesisstates that w_izcan be represented as follows:

w_iz¼X^N

j¼1

zjwij ð1Þ

whereNdenotes the total number of self pMHC species that can bind to the TCR at hand (this number is assumed to be the same for all TCR clonotypes). In the present paper it will be assumed thatw_ijis a Bernoulli variate:

For a given clonotypeithe values ofw_ijfor the various pMHC speciesjdetermine the identity of the clonotype.

The restriction ofw_ijto two values (0 and 1) is motivated in Appendix A. The quantitiesw_izandw_ijare dimension- less, and have been scaled relative to the maximum value attainable by the TCR signal (see Appendix A).

Letmdenote the probability that a randomly selected TCR recognises a randomly selected pMHC species; thus, m¼P{w_ij¼1} for a randomly chosen pair (i,j). The parameter m represents the inherent degeneracy of TCR/pMHC recognition (Mason, 1998; Borghans et al., 1999). Its reciprocal 1/m is the inherent specificity, and equals the average number of clonotypes that must be examined before one is found that recognises the pMHC species at hand. Pathogenic and normal autoantigens are not formally distinguished: the same parametermdenotes both the probability that a T cell will recognise a foreign antigen and the probability that an autoantigen will be recognised by a positively selected thymocyte before negative selection.

It may be assumed thatm!1: even though TCR recognition degeneracy is considerable (Mason, 1998), any given TCR will be unable to interact productively with the vast majority of pMHC species with which it can interact;mis typically estimated to lie in the range 10²⁵– 10²⁴(Gavin and Bevan, 1995; Butz and Bevan, 1998; Mason, 1998); for certain antigens the frequency of T cells in the naı¨ve repertoire that recognise the antigen may be higher: for instance, it was found that over 1 in 2500 CD8^þT cells are specific for the Melan-A/MART-1 autoantigen in HLA-A2 individuals (Pittetet al., 1999), and the frequency of specific precursors in the repertoire to antigens such as cytomegalovirus may be higher still (Oelkeet al., 2003).

Thethreshold hypothesisstates that the T cell becomes activated when the TCR signal is greater than some threshold value. In fact, the T cell may be capable of various responses, some of which may be more readily evoked than others (Valituttiet al., 1996; Itoh and Germain, 1997). Accordingly, the T cell may be assumed to have various different thres- hold values, each corresponding to a particular response.

Two important examples of such responses are (i) the naı¨ve T cell’s decision to commit to differentiation and

proliferation, which happens whenw_izexceedsw_act(Viola and Lanzavecchia, 1996; Lanzavecchia and Sallusto, 2000) and (ii) the thymocyte’s entry into apoptosis, which happens whenwiz exceeds the selection threshold wthy (Bouneaud et al., 2000; Savage and Davis, 2001).

Component Representation of Self

The number N of self pMHC species that can interact with a given TCR is very large (Bevan et al., 1994;

Mason, 2001), and this fact can be exploited to give large deviations estimates for the repertoire structure. Thus, it is essential that antigen presentation be described in a way that allows the limitN! 1to be taken. To this end, the self pMHC species will be partitioned intoK,1self- presentation components, such that all pMHC species belonging to a given component have two characteristics in common: their frequency of occurring in an APP and their typical presentation level.

The component approach allows a compact represen- tation of fluctuations in TCR signalling due to self. The essential idea is that each component collects self pMHC species that share their ubiquity and mean presentation level. This section first explains the general concept, and then details the particular model for self-fluctuations used in the present paper.

The Concept of Self Presentation Components

A given T cell will in general not register the same TCR signal during subsequent encounters with different APCs, because the ensemble of presented pMHC species will differ; if the encounters are sufficiently far apart in time, the same will even be true of the same APC. Such variations in the APP occur because not every self peptide occurs in every APP, even within a defined class of APCs such as the negatively-selecting cells. Moreover, the presentation level of a peptide will vary as the total numbers of peptides present at non-zero presentation levels also fluctuates between APPs. The idea behind the component approach introduced in this paper is to characterise these various kinds of fluctuations in a non- degenerate manner as N! 1, by representing the TCR signalw_izin terms of presentation components rather than the pMHC species themselves.

Every self peptide speciesj has a specific ubiquity u_j relative to a given class of APCs: u_j[[0,1] represents the probability that the peptide is presented by a randomly chosen APC from this class. A pMHC species present in every APP will have ubiquity one, while w_ij¼

0 when TCR i does not productively recognise pMHC species j 1 when TCR i productively recognises pMHC species j:

(

ð2Þ

a species presented very rarely will have a ubiquity close to zero. Finally, a peptide never expressed by the APCs at hand will have ubiquity zero for that class of APCs. The numberNrefers to all autoantigens that can interact with a given TCR, with or without productive recognition.

One would therefore expect many ubiquities to be identically zero for any particular class of APCs, although professional APCs may be more versatile than many other cell types in this respect (Steinman et al., 2000). Besides their ubiquities, pMHCs also differ with respect to their presentation levels (Hunt et al., 1992;

Morgan et al., 1999; Reayet al., 2000). Thus, every self peptide is furthermore characterised by its average presentation level zj: A pMHC species j is thus characterised by two parameters: its ubiquity u_j and its average relative presentation levelzj:

The component approach distributes the N self peptides among K components so that peptides that belong to the same component have the same ubiquity and average presentation level. Thus, these parameters can be indexed by components k instead of peptides j, such that u_j¼u_kand zj¼zk whenever pMHC speciesj belongs to component k. While a pMHC species will generally have to be assigned to a component whose ðuk;zkÞ does not exactly match ðuj;zjÞ; the number of componentsKcan always be increased for a finer-grained classification.

Two classes of professional antigen-presenting cells will be considered in this paper: the negatively selecting cells which mediate negative selection and the pro- fessional antigen-presenting cells (pAPCs) in the secondary lymphoid tissues which activate naı¨ve T cells. The latter will be distinguished by a prime: u⁰_k;z⁰_k. For two peptides to belong to the same self presentation component, it is only required that they share ubiquity and mean presentation level in both classes, while it is not required that these values be the same in the two classes: where kdenotes a component andj₁and j₂two pMHC species,

j₁[k and j₂[k iff

uj₁ ¼uj₂¼uk & zj₁¼zj₂ ¼zk &

u⁰_j

1 ¼u⁰_j

2¼u⁰_k & z⁰_j1 ¼z⁰_j2 ¼z⁰_k

while not necessarily uk¼u⁰kor z⁰_k¼z⁰_k:Consequently, two distinct components k and ‘ may have the thymic statistics in common ðu_k¼u_‘ and zk¼z‘Þ but owe their status as distinct components to peripheral differences (e.g.u_k⁰–u⁰‘). The ubiquities for the various components are collected in aK-vectoru(for the class of negatively selecting cells) oru⁰(for the class of pAPCs in the secondary lymphoid tissues).

In what follows it will be useful to focus on the fraction of MHC molecules that present a pMHC belonging to componentk. This fraction is the component presentation

levelr_k:

rk def¼

N!1lim

j[Kk

Xzj ð3Þ

whereKkis a subset of {1,2,. . .,N} such thatj[Kkiffj belongs to component k. The component presentation levels are collected in a K-vector r with PK

k¼1rk¼1;

inner products of K-vectors will often be written more compactly, thus: k1;rl ;PK

k¼1rk: The class of nega- tively selecting thymic stroma cells is then parametrised by ubiquities u and component presentation levels r, which together describe the statistical fluctuations of self antigen presentation within that class. Similarly, the class of pAPCs in the secondary lymphoid tissues is characterised by its own pMHC ubiquities u⁰ and component presentation levelsr⁰.

The Ubiquity – Propensity Model

The concept ofpresentation propensitywas introduced by van den Berg et al. (2001) to describe APP variations between APCs with varying protein expression patterns.

The relative presentation level of a peptide depends on a number of factors: for MHC class I presentation, these factors include whether the peptide is expressed in the APC; the level at which it is expressed (Jardetzkyet al., 1991); the likelihood that the peptide is properly processed (protein degradation in the proteasome, transport to the intracellular compartment containing the MHC molecules (Stevanovic´ and Schild, 1999)) and its affinity for the MHC binding cleft, which determines its ability to compete with other peptides. The propensity model combines these factors into a single, aggregate propensity measure r$0: The key idea is that the ratio between presentation levels of every pair of expressed peptidesjandj⁰is given by the ratio of their propensities, z_j/z_j⁰¼r_j/r_j⁰wherer_jandr_j⁰are the propensities of the two peptides. Thus, peptides with equal propensities have equal presentation levels, when both are expressed. Class II presentation is modelled in a similar way, although the propensity now reflects the properties of the endocytic processing pathway, and the presence of a peptide in the APP is governed by the protein content of the endocytic vesicles rather than expression by the APC itself.

On the propensity model, the relative presentation level zjof pMHC speciesjis found by dividing the propensity r_j of the peptide by the sum of the propensities of all peptides that are present in the MHC loading compart- ment. If I_jz is an independent Bernoulli variate which takes on the value 1 with probabilityu_kwhenjbelongs to componentk, then

zj¼I_jz rj

r_Tz with r_Tz ^def¼ X^N

j¼1

r_jI_jz; ð4Þ

and N independent realisations of Bernoulli variates I_jz— one for each self pMHC species—are required to specify a single APPz.

Letp_kdenote the probability that a randomly selected self pMHC belongs to componentk, so that there areNk

def¼pkN pMHC species in thekth componentðjKkj ¼NkÞ;and the distribution of self pMHC species over the components is represented by aK-vectorpwithk1;pl¼1:From the fact that lim_N!1r_Tz=N¼PK

k¼1r_ku_kpk where r_k denotes the propensity shared by all pMHCsj[Kk, it follows that the component presentation level is given by

rk¼ r_ku_kpk

PK

‘¼1r‘u‘p‘

:

The mean presentation level for a pMHC species belonging to componentkcan now be expressed as

zk¼ r_k NPK

‘¼1r‘u‘p‘

¼ rk

ukpkN; ð5Þ and the TCR signal due to self, Eq. (1), can be rewritten as a sum over the components (with Eqs. (4) and (5)):

w_iz¼X^N

j¼1

I_jz rj

r_Tzwij¼X^K

k¼1

rk

r_{Tz j[K}

k

XI_jzwij¼X^K

j¼1

rk

r_Tzn^_ikz

wheren^_ikz^def¼P

j[KkI_jzw_ij denotes the number of pMHC speciesjthat belong to componentkand are (i) presented in APPz(I_jz¼1) and (ii) are productively recognised by the TCR of clonotypei (w_ij¼1). For large N, the TCR signal due to self becomes:

N!1limw_iz¼X^K

k¼1

zkn^_ikz¼X^K

k¼1

rk

ukpk

^ n_ikz

N ð6Þ

which is well-defined as the quantity n^_ikz=N does not become degenerate in the limit N! 1. Stochastic fluctuations of the recognised pMHC species {^n_ikz}^K_k¼1 are partly due to differences between clonotypes i, and partly due to differences between APPsz. In other words, there is both an across-clonotype and a within-clonotype contribution to the fluctuations ofn^_ikz:Negative selection reshapes the T cell population through differential probabilities of survival, but cannot alter the autorecogni- tion properties of any given thymocyte; all that negative selection can do is to delete certain clonotypes.

Fluctuations across and within Clonotypes

The basic technical difficulty in the analysis of T cell tolerance is the fact that the T cell repertoire is a

“distribution of distributions”: each clonotype has its own across-APP distribution of self stimulation, and the repertoire as a whole is a collection of such distributions.

The component-based approach allows the introduction of a law (distribution) specific for each clonotype.

Variability across the APPs for a given clonotype is governed by the law of that particular clonotype.

By contrast, variability over the clonotypes is represented by the ensemble of the clonotype laws, which models the statistical structure of the TCR repertoire, and which is modified by negative selection.

Recognition Frequencies: The Clonotype Law

Let N^ik denote the number of self pMHC species belonging to a given componentkthat are recognised by a TCR of clonotypei:

N^ik def

¼

j[Kk

Xwij: ð7Þ

While fixed for a given combination of clonotypeiand component k, the number N^ik varies randomly between clonotypes. Prior to negative selection, N^ik follows a binomial distribution:

PðN^_ik¼xÞ ¼ Nk

x

!

m^xð12mÞ^Nk2x ð8Þ

provided that recognition of pMHC species is statistically independent. This is a reasonable assumption in view of the fact that degeneracy in TCR/pMHC recognition is polyspecific, with unfocused cross-reactivity (Hagerty and Allen, 1995; Ignatowiczet al., 1997; Mazzaet al., 1998;

Andersonet al., 2000).

A TCR clonotypeiis fully specified by the numbersN^ik

of recognised pMHC species in the various componentsk.

Normalising these numbers to their respective maximum values, one obtains theclonotype lawL^_i:

L^i def

¼

{

}

which is an element of the setLNdefined by LN

def¼

{

}

whereN ^def¼ {0;N1}£· · ·£{0;NK}:Thekth element of the law L^ik¼N^ik=N_k is the clonotype’s autorecognition frequency for pMHC species belonging to componentk.

Thus, the component approach represents a T cell’s autorecognition entirely by the law of the clone to which it belongs. It will often be convenient to refer more briefly to a T cell’s lawL:^

Since each of the N^ik recognised autoantigens has a probabilityu_kof being present at a non-zero presentation level in a random APP, the average TCR signal due to self of clonotypei, the clonotype mean w_i; is PK

k¼1zku_kN^_ik; which can be written succinctly in terms of the component presentation levels and the clonotype law:

w_i¼kr;L^_il: ð9Þ Thus, a T cell with autorecognition frequencies L^ experiences an average TCR signal kr; Ll^ from self peptides it encounters during negative selection in the thymus, and kr⁰; Ll^ is the average TCR signal due to autorecognition registered in the secondary lymphoid tissues. A large deviations rate function forwiis derived in Appendix B.1.

Prior to negative selection, the expectation over the clonotypesiof the autorecognition frequency ismfor all components k, and thus the expectation over the clonotypes of wi is m as well. One effect of negative selection is to introduce correlations between the elements of the law L^ so that they can no longer be treated as independent. For instance, if it is known for some post- selection T cell with lawL^that one of the elements ofL^is larger thanm, there will be a reduced probability, relative to the unconditional value, that another element also exceedsm.

Antigen Presentation Fluctuations

Differences between clonotypes are expressed by differ- ences between the corresponding clonotype laws L^i: Their distribution reflects the statistical structure of the repertoire, and it is this distribution that negative selection acts upon. Within a given clonotype, statistical fluctuations remain; these are due to differences in APPs, which cause the TCR signal wiz to fluctuate about the clonotype meanw_i. To characterise the within-clonotype fluctuations of the TCR signal across APPs, observe that component k contains N_k self pMHC species, out of which a number n_kz^def¼P

j[KkI_jz have a non-zero presentation level in APP z. Out of these n_kz presented pMHC species, a number n^_ikz¼P

j[KkI_jzwij

is recognised by a TCR of clonotype i. A standard argument shows that n^_ikz follows the hypergeometric distribution:

Pð^n_ikz¼xÞ ¼ N^ik

x

! Nk2N^ik

n_kz2x 0

@

1 A

N_k n_kz

! ð10Þ

on the assumption that peptides are independently presented. The description of the TCR signal w_izin the limit N! 1, Eq. (6), requires that the random variable

^

n_ikz=Nremains non-degenerate in this limit; this is shown in Appendix B.2, where a large deviations rate function for n^_ikz=Nkis derived.

STATISTICS OF AUTOREACTIVITY FOLLOWING NEGATIVE SELECTION

This section presents three main results. The first result is a characterisation of the post-selection statistics of clonotype laws over the repertoire, in terms of a large deviations rate function. Then follows a second-order approximation of this rate function for an important class of clonotype statistics. The third result relates thymic presentation statistics to the role of negative selection in improving immune efficacy.

The Post-selection Large Deviations Rate Function LetP_survdenote the probability that a thymocyte, chosen at random from among the thymocytes restricted to a given single MHC isoform, will appear in the mature naı¨ve repertoire. This probabilityPsurvis not to be confused with the overall probability that a thymocyte will be allowed to mature. The latter includes both positive and negative selection, and its magnitude may largely reflect positive selection (Surh and Sprent, 1994; Merkenschlageret al., 1997; Laufer et al., 1999; Sebza et al., 1999). In other words,P_survdenotes a thymocyte’s probability of surviving into the mature repertoire conditional on being positively selected; it is given by

Psurv¼

n[LN

XP{survivaljL¼^ n}P{L¼^ njpre-selection}:ð11Þ

In this formula and those that follow, the term

“selection” refers to negative selection only. The post- selection structure of the TCR repertoire can be discussed in terms of the probability that a mature T cell that has survived negative selection has a given lawL^i:Bayes’ rule relates this probability to pre-selection probabilities:

P{L^ ¼njpost-selection}¼

P{L^ ¼njpre-selection}P{survivaljL^ ¼n}=Psurv ð12Þ where n[LN and i is the T cell’s clonotype. The conditional survival probability is given by

P{survivaljL^ ¼n}¼

P{w_iz,wthyjL^ ¼n}m

ð13Þ where w_thy is the thymic selection threshold of the thymocyte at hand andmdenotes the number of antigen presentation events during which the T cell will undergo deletion if it registers a TCR signal greater than w_thy (Savage and Davis, 2001). Thus, m is the number of rounds of negative selection. Experimental estimates of mare not available, although there are some indications that thymocytes remain sufficiently long in the thymus to allow for dozens or perhaps even hundreds of rounds of negative selection (Scollay and Godfrey, 1995); this should however be set against the possibility that m is only of order 1, as may be the case for positive selection (Merkenschlager et al., 1994; Merkenschlager, 1996).

Given a post-selection T cell which has successfully undergonemrounds of negative selection, the probability that this T cell’s law will belong to an arbitrary open set G,[0,1]^K can be estimated from the following large deviations result (Dembo and Zeitouni, 1998):

N!1lim 1

N lnP{L^ [Gjpost-selection}^def¼I_G¼ 2n[Ginf HðnjmÞ þm

w:k1;wl#wthy

inf {kp;Iðw;n;u;rÞl}2Isurv

ð14Þ

where IG is the post-selection rate function associated with the set G. Equation (14) shows how to find this rate function from the rate functions associated with fluctuations over clonotypes and over APPs: H(njm) denotes the relative entropy ofn, defined in appendix B.1;

I(w;n,u,r) is aK-vector of large deviations rate functions for the various components, defined in appendix B.2, and

Isurv def

¼ N!1lim2ln {Psurv}=N: ð15Þ The evaluation of the infimum in Eq. (14) is somewhat involved, and is discussed in detail in appendix B.3.

Approximate Statistics of the Post-selection Repertoire All the information regarding the post-selection TCR repertoire is contained in the large deviations rate function I_G. To understand the behaviour of this rate function one can consider its Taylor series aboutm. This section gives leading-order estimates which afford insight into the relationship between the selection parameters and the presentation statistics of the negatively selecting cells.

The Survival Probability

The stringency of negative selection can be expressed as the probabilityP_survthat a thymocyte, chosen at random from among those thymocytes that are restricted to one MHC isoform, will appear in the mature naı¨ve repertoire.

One intuitively expects this survival probability P_surv to decrease with an increasing number of rounds of selection mand to increase with the selection thresholdw_thy. This is borne out by the following estimates:

Psurv<

1 wthy.m

exp 2N_rmþ^m_m

crit

ðwthy2mÞ² 2m

n o

w_thy,m 8>

<

>: ð16Þ

where Nr def

¼ NðPK

k¼1r²_k=pkÞ²¹ and mcrit denotes the critical number of presentation rounds, itself defined in terms of theeffective mean ubiquityu :

mcrit def

¼ 12u

u where u ^def¼ PK

k¼1r²_k=pk

PK

k¼1r²_k=ðpku_kÞ: ð17Þ The effective mean ubiquity represents an average pMHC ubiquity in which presentation levels are taken into account, as can be seen more clearly from the following approximate formula in terms of data describing a representative APPz:

u <

PN j¼1z²_juj

PN j¼1z²_j ^:

This shows howumay be determined empirically from antigen presentation data on the negatively selecting cells.

Clearly, u is near 1 when most of the dominant, high z;

self pMHC species are present in almost every APP of the negatively selecting cells, whileuis almost zero when all pMHCs have a low probability of being presented on any given negatively selecting cell.

The effect of the number of presentation rounds depends on the typical ubiquity of the self pMHC species.

In particular, when all pMHCs are always presentðu¼1Þ;

the across-APP variations vanish and a given thymocyte will register the same TCR signal on every conjugation, which means that m_crit¼0 and the value of m is immaterial as long as it is at least 1. On the other hand, when the typical ubiquity is very small,m_critwill be very large, and the survival probability will continue to decrease withm(form&m_crit).

The number Nrcan be determined from experimental data by means of the approximation

N_r<ndiv

=

Heren_divdenotes the presentation diversity, defined for a particular APPzas follows:

ndiv ¼1.X^N

j¼1

z²_j: ð18Þ

This presentation diversity is at least 1, a lower bound which is attained when a single pMHC species j⁰ dominates with presentation level zj⁰¼1. Furthermore, n_div is at most the number of distinct pMHC species present in an APP, which isNkp,ul#N; this upper bound is attained when all pMHC species present have the same presentation level.

The Clonotype Law after Negative Selection

The statistical distribution of clonotype laws among the post-selection T cells that make up the mature repertoire is of fundamental importance to immune response efficacy, tolerance and autoimmunity. This distribution can be characterised in terms of the distributions of a family of clonotype statistics of the formka; Ll^ wherealies in the simplexSdefined by

S^def¼ x[½0;1^K :X^K

k¼1

xk¼1

( )

: ð19Þ

Such a weighing vectoracan be used to represent the target of negative selection. This motivates the study of the across-repertoire distribution of statistics of the form ka;Ll^ before and after negative selection. In particular, if a vectora~[Scan be found such that clonotypes with a high value of ka~;Ll^ ought to be preferentially removed from the repertoire, the across-repertoire distribution of this statistic ka~;Ll^ should be sharply truncated by the action of thymic selection. In this sense,a~represents the target of negative selection. Claim 1 below relates this target to thymic presentation of autoantigens.

Clonotype statistics of the form ka;Ll^ generalise the clonotype mean wi (since wi¼kr;L^il and r[S). The post-selection statistical structure of the TCR repertoire may be studied by considering the family of distributions of the statisticka;Ll^ parametrised bya[S, whereL^ is the law of a T cell chosen at random from the mature naı¨ve repertoire. Approximations for this family of distributions follow shortly in Eqs. (22) – (24) below. These equations constitute a representation of the statistical structure of the repertoire after negative selection. The immunological interest is in exhibiting the roles played by the thymic presentation parameters and the selection parametersw_thy and m, and in establishing the principle of maximum selection efficiency (see “The efficiency of negative selection” section below).

Preliminary to the statement of the approximations, a few further compound parameters are defined. The following two coefficients relate the thymic presentation parameterrto the weighing vectora:

1_a ^def¼

PK

k¼1akrk=pk

²

PK

k¼1a²_k=pk

PK

k¼1r²_k=pk

and

r_a ^def¼ PK

k¼1a²_k=pk

PK k¼1r²_k=pk

: ð20Þ

Finally, letN_a^def¼NðPK

k¼1a²_k=pkÞ²¹¼N_r=r_aand let

vcritdef

¼

mþpffiffiffiffiffiffiffiffiffiffir_a1_a

ðw_thy2mÞ=ð1þm_crit=mÞ forw_thy#m mþ ffiffiffiffiffiffiffiffiffiffiffiffi

r_a=1_a

p ðw_thy2mÞ forwthy.m: 8<

:

ð21Þ Large deviations theory only describes the effect of m$0 rounds of negative selection for ka;Ll^ .vcrit: In the case wherewthy#m, the following approximation obtains: for v#v_crit, P{ka;Ll^ .vjm rounds}<1 whereas forv.vcrit,

P{ka;Ll.^ vjmrounds}

<exp 8<

: 2N_a

2m 0

@ðv2mÞ²þ ffiffiffiffiffi 1_a

p ðv2mÞ2 ffiffiffiffiffira

p ðw_thy2mÞ

2

121_aþm_crit=m

hskip38pt2r_aðwthy2mÞ² 1þm_crit=m

1 A

:

When w_thy.m, the approximation is slightly more complicated: forv#m,P{ka;Ll^ .vjm rounds}<1;

next, form,v#vcrit,

P{ka;Ll^ .vjm rounds}<exp 2N_aðv2mÞ² 2m

ð23Þ

and, finally, forv.v_crit, P{ka;Ll^ .vjm rounds}

<exp (2N_a

2m ^ðv2mÞ²

þ ffiffiffiffiffi 1_a

p ðv2mÞ2pffiffiffiffiffir_a

ðw_thy2mÞ

2

121_aþmcrit=m

!) : ð24Þ

The pre-selection approximations are obtained for m¼0 in these expressions: in that case, vcrit¼m, and P{ka;Ll^ .vjm rounds}<1 forv#m, while Eq. (23) holds forv.m.

Post-selection Autorecognition Frequencies

Prior to negative selection, the average autorecognition frequency L^k among the thymocytes is m for every componentk. Negative selection reduces this average by rendering high autorecognition frequencies less likely.

The autorecognition frequency is a statistic of the form ka;Ll^ witha¼1k;that is,a_k¼1 anda‘¼0 for‘–k:

L^k¼k1k;Ll:^

In the case where w_thy.m, large deviations in autorecognition frequencies after m rounds of negative selection are described by

P{L^k.vjm rounds}

<exp 2pkN

2m ^ðv2mÞ²þ 11_kðv2vcritÞ² 121_1kþmcrit=m

forv.v_crit, where vcrit¼mþwthy2m

rk

and 11_k ¼ r²_k=pk

PK

‘¼1r²_‘=p‘

:

Together, these expressions indicate that a sharp truncation atL^k¼wthycan be achieved only if component k dominates thymic presentation, with r_k close to 1.

Otherwise, the truncating effect is imperfect and becomes prominent only at higher values of the autorecognition frequency. If the component is not prominent at all during thymic selection, the post-selection distribution will hardly differ from the pre-selection distribution. For such a component, self-nonself discrimination fails, and its self pMHC species are effectively foreign. Indeed, thymic expression of autoantigens has been found to correlate with resistance to autoimmune disease (Egwuaguet al., 1997).

Similar conclusions apply in the case where w_thy,m. These observations are generalised in the next section.

The Efficiency of Negative Selection

The extent to which negative selection in the thymus has modified the TCR repertoire can be gauged by studying

the post-selection distribution ofka;Ll^ :varyingawithin the simplex S, one obtains a family of distributions P{ka;Ll^ .v} which probe the post-selection structure of the TCR repertoire. Negative selection reduces the probability that ka;Ll^ exceeds a givenv. The extent of the reduction depends on bothaandr, and is maximised whenaandrcoincide, as is expressed by the following claim.

Claim 1 (Maximum Efficiency of Negative Selec- tion) Given thymic presentation statistics u, a fixed selection parameter w_thy, a fixed but sufficiently large number of self peptidesN, a weighing vectora[S, and v.w_thywithvsufficiently close tom, the probability

P{ka;Ll^ .vjafter m rounds of negative selection}

can be made arbitrarily small by choosingmsufficiently large and the thymic presentation vector r sufficiently close toa.

On the approximations of Eqs. (22) and (24), the large deviations rate function can be made arbitrarily large only if the quantity

121_aþmcrit=m

is made arbitrarily small (this follows from the fact that bothra and1a have finite upper and lower bounds asr varies over S, with a fixed). This is shown in Fig. 1.

By Ho¨lder’s inequality,

1_a#1

with equality only whenr¼a. Thus, 121_a is smaller than a given positive bound iffris sufficiently close toa, and Claim 1 follows. The claim must be restricted to v.w_thy, since vcrit!w_thy when r!a. The present proof is based on a second-order approximation of a large deviations estimate, which imposes the following technical restrictions: (i) the number of self peptides N

has to be large enough for the large deviations-based approximation to be relevant; and (ii)w_thyandvhave to be sufficiently close tomto ensure that the large deviations function is dominated by its second-order term.

In general, the value of wthy must be expected to vary among thymocytes, and it may even change for a given thymocyte as it matures. In view of these variations, it is significant that Claim 1 does not hinge on the value of w_thy but rather on the characteristics m and r, which are governed by thymic architecture and differentiation of the thymic stroma (Laufer et al., 1999; Sebza et al., 1999).

According to Claim 1 the thymic presentation vectorr encodes, effectively, the target of negative selection.

The relationship between thymic presentation and immune efficacy only makes sense in the light of the statistical fluctuations of peripheral self presentation. This is explained in the next section by means of two illustrations of how Claim 1 can be applied.

IDENTIFYING THE TARGET OF NEGATIVE SELECTION

The efficiency result, Claim 1, implies that the thymic presentation vectorris an objective representation of the target of negative selection. In particular, among all clonal statistics of the formka;Ll^ with a[S, the one that is most efficiently selected against iskr;Ll:^ To understand the immunological significance of the thymic presentation vector, it is useful to apply the efficiency result in the opposite direction: that is, formulate a hypothesis concerning the target of negative selection, encode it as a weighing vectora[S, and compare this to the actual thymic presentationr. While this is, in principle, a feasible empirical programme, difficulties may arise given experimental limitations in assessing presentation levels and ubiquities of a large sample of self peptides, among

FIGURE 1 Efficiency of negative selection. The probability that the clonal statisticka;Ll^ exceedsv, plotted as a function ofv/m. Dotted line indicates the pre-selection case. Curves are from left to right for decreasing1a, a parameter which expresses how wellaand the thymic presentation vectorr are matched;1a¼1, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1. Top panels: the number of rounds of negative selectionmequalsmcritdef¼ ð12uÞ= uwhere

uis the effective mean ubiquity of thymic presentation; bottom panels: limiting case wheremis much larger thanmcrit. Left panels:wthy¼1.1m; right panels:wthy¼0.5m. All panels:ra¼1 Nadef¼NðPK

k¼1a²_k=pkÞ²¹

:

both the classes of negatively selecting cells and secondary lymphoid pAPCs.

The presentation statistics of the professional APCs (pAPCs) in the secondary lymphoid tissues are indicated with a prime:r⁰,u⁰. Derived statistics such as presentation diversityn_div⁰ and mean ubiquityu⁰are important because they can be determined from experimental data.

Approximate formulæ are:

u⁰<

PN j¼1z⁰²_ju⁰_j PN

j¼1z⁰²_j ^{and n}

0

div< 1 PN

j¼1z⁰²_j ^;

to be determined from data on a representative APP z⁰ from a secondary lymphoid pAPC.

In the present section, the programme of identifying the target of negative selection will be illustrated by an application to two hypotheses concerning the target of negative selection. Whereas the first of these two hypotheses (negative selecting is means-directed) probably reflects a consensus among immunologists, the second (negative selection is variance-directed) is perhaps more controversial.

Means- versus Variance-directed Negative Selection Two natural parameters of a mature recirculating T cell’s autoreactivity are the average and the variance of the TCR signal due to self registered by the T cell as it interacts with pAPCs in secondary lymphoid tissues throughout the body. While both parameters are prima faciea candidate target for central tolerance, the post-selection structure of the T cell repertoire would be markedly different, as shown schematically in Fig. 2.

Means-directed Negative Selection

A mature recirculating T cell with autorecognition frequencies L^ experiences an average TCR signalkr⁰;Ll^ from self peptides on the pAPCs it encounters. A natural assumption is that negative selection acts to remove the T cells with the highest means kr⁰;Ll:^ This assumption identifies the weighing vector a with the peripheral presentation vectorr⁰. The maximum efficiency principle, Claim 1, then impliesr¼r⁰for the thymic presentation vector. The approximations presented in the previous section then take on a much simpler form: forw_thy#m and v.vcrit¼mþ ðwthy2mÞm=ðmþmcritÞ; Eq. (22) becomes:

P{kr⁰;Ll^ .vjm rounds}

<exp 2n⁰_div=u⁰ 2m

ðv2mÞ²

þ m mcrit

ðv2wthyÞ²2ðw_thy2mÞ² 1þmcrit=m

; ð25Þ

whereas forw_thy.mandv.w_thy, Eq. (24) becomes:

P{kr⁰;Ll^ .vjmrounds}

<exp 2n⁰_div=u⁰

2m ^ðv2mÞ²þ m

m_critðv2w_thyÞ²

: ð26Þ

These expressions clearly show that the number of presentation roundsmmust be of orderm_critor larger to obtain a sizable post-selection improvement of the TCR repertoire, as is shown in Fig. 3. The critical number of presentation roundsm_critdecreases with increasing thymic effective mean ubiquityu: However,r¼r⁰implies that ndiv=u¼n⁰_div=u⁰:Thus, if the mean ubiquity in the thymus

u is higher than the peripheral value u⁰; this must be compensated for by a proportionally greater presentation diversity ndiv on the negatively selecting cells. This, in turn, would require high MHC counts on the negatively selecting cells to safeguard signalling fidelity (see van den Berg et al., 2001 and van den Berg and Rand, 2003 for detailed arguments).

Variance-directed Negative Selection

An alternative hypothesis is that negative selection is directed against the variance ofw_iz, the TCR signal across the APPs on professional APCs in the secondary lymphoid tissues. Thus, thymic selection particularly targets those clonotypes whose across-APP variance in the periphery is large. The question then becomes how thymic

FIGURE 2 Two hypotheses concerning the target of negative selection.

The probability that the TCR signal due to self exceedsvas peripheral APCs vary, plotted as a function ofvfor four representative clonotypes:

(a) a clone with a low mean and small variance; (b) a clone with a low mean and large variance; (c) a clone with a large mean and small variance; (d) a clone with a large mean and large variance. Top panel:

situation prior to negative selection. Middle panel: situation after selection targeted at clonotype means. Bottom panel: situation after selection targeted at clonotype variances.